Lorenzo Demenia - Esercizio 2
Questo documento va caricato in Moodle come cognome-nome-esX.Rmd dove X è il numero dell’esercizio (1, 2, 3 o 4).
1 Soluzione quesito 1
scrivere qui la soluzione del primo quesito
2 Soluzione quesito 2
scrivere qui la soluzione del secondo quesito
3 Soluzione quesito 3
scrivere qui la soluzione del terzo quesito (sempre che ve ne siano tre, altrimenti cancellare questa sezione)
4 Prototipazione Regressione
5 Inserimento dei dati e Analizziamo
- Leggiamo di dati e Visualizziamo:
dataset <- read.csv('efficienza.csv')
# x_name <- processi
# y_name <- dimensione
plot(processi ~ dimensione, data = dataset)
summary(dataset)## processi dimensione
## Min. : 1.00 Min. : 6.0
## 1st Qu.: 12.00 1st Qu.: 8.0
## Median : 26.00 Median :14.0
## Mean : 35.52 Mean :13.1
## 3rd Qu.: 48.00 3rd Qu.:17.0
## Max. :100.00 Max. :20.0- Definiamo le osservazioni anomale:
outliers_range <- 80
outliers <- which(dataset$processi > outliers_range) # processi è la x_name
summary(outliers)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 8.00 8.75 13.50 16.00 20.75 29.006 Stimiamo il modello
6.1 Stima della retta di regressione:
- Normale
mod <- lm(processi ~ dimensione, data = dataset)
plot(processi ~ dimensione , data=dataset)
curve(coef(mod)[1] + coef(mod)[2] * x + coef(mod)[3] * x ^ 2, from = min(dataset$processi), to = max(dataset$processi), add = TRUE, col = "blue")
abline(mod, col = 'red')
summary(mod)##
## Call:
## lm(formula = processi ~ dimensione, data = dataset)
##
## Residuals:
## Min 1Q Median 3Q Max
## -26.392 -19.051 -11.370 4.014 69.289
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 57.261 16.551 3.460 0.00181 **
## dimensione -1.659 1.199 -1.384 0.17783
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 27.94 on 27 degrees of freedom
## Multiple R-squared: 0.0662, Adjusted R-squared: 0.03162
## F-statistic: 1.914 on 1 and 27 DF, p-value: 0.1778- Analisi con i residui:
par(mfrow = c(1,2))
plot(residuals(mod) ~ dimensione, data=dataset)
abline(h=0, col = "red")
qqPlot(residuals(mod))
## [1] 8 9 par(mfrow = c(1,1))- Analisi senza i residui:
mod2 <- lm(processi ~ dimensione , data = dataset, subset=-outliers)
plot(processi ~ dimensione , data=dataset, subset= -outliers)
summary(mod2)##
## Call:
## lm(formula = processi ~ dimensione, data = dataset, subset = -outliers)
##
## Residuals:
## Min 1Q Median 3Q Max
## -11.9634 -4.6211 0.5299 4.0233 13.1877
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 74.6613 4.2659 17.50 8.67e-15 ***
## dimensione -3.8356 0.3217 -11.92 2.51e-11 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 7.034 on 23 degrees of freedom
## Multiple R-squared: 0.8608, Adjusted R-squared: 0.8547
## F-statistic: 142.2 on 1 and 23 DF, p-value: 2.508e-11 plot(processi ~ dimensione, data = dataset)
curve(coef(mod)[1] + coef(mod)[2] * x + coef(mod)[3] * x ^ 2, from = min(dataset$processi), to = max(dataset$processi), add = TRUE, col = "blue")
curve(coef(mod2)[1] + coef(mod2)[2] * x + coef(mod2)[3] * x ^ 2, from = min(dataset$processi), to = max(dataset$processi), add = TRUE, col = "green")
abline(mod, col = 'red')
par(mfrow = c(1,2))
plot(residuals(mod2) ~ dimensione[-outliers], data=dataset)
abline(h=0, col = "red")
qqPlot(residuals(mod2))
## [1] 3 1 par(mfrow = c(1,1))6.2 Stima Quadratica:
mod_quad <- lm(processi ~ dimensione + I(dimensione ^ 2), data = dataset)
plot(processi ~ dimensione + I(dimensione ^ 2), data=dataset)
curve(coef(mod_quad)[1] + coef(mod_quad)[2] * x + coef(mod_quad)[3] * x ^ 2, from = min(dataset$processi), to = max(dataset$processi), add = TRUE, col = "blue")
summary(mod_quad)##
## Call:
## lm(formula = processi ~ dimensione + I(dimensione^2), data = dataset)
##
## Residuals:
## Min 1Q Median 3Q Max
## -27.930 -19.101 -9.357 3.668 70.423
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 78.5531 54.4679 1.442 0.161
## dimensione -5.4942 9.4102 -0.584 0.564
## I(dimensione^2) 0.1521 0.3700 0.411 0.684
##
## Residual standard error: 28.38 on 26 degrees of freedom
## Multiple R-squared: 0.07223, Adjusted R-squared: 0.0008619
## F-statistic: 1.012 on 2 and 26 DF, p-value: 0.3773- Analisi con i residui:
par(mfrow = c(1,2))
plot(residuals(mod_quad) ~ dimensione, data=dataset)
abline(h=0, col = "red")
qqPlot(residuals(mod_quad))
## [1] 8 9 par(mfrow = c(1,1))- Analisi senza i residui:
mod_quad_2 <- lm(processi ~ dimensione + I(dimensione ^ 2), data = dataset, subset=-outliers)
plot(processi ~ dimensione + I(dimensione ^ 2), data=dataset, subset= -outliers)
summary(mod_quad_2)##
## Call:
## lm(formula = processi ~ dimensione + I(dimensione^2), data = dataset,
## subset = -outliers)
##
## Residuals:
## Min 1Q Median 3Q Max
## -14.8440 -3.4158 0.5015 4.0487 10.5842
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 104.33360 12.36306 8.439 2.4e-08 ***
## dimensione -9.18934 2.13998 -4.294 0.000294 ***
## I(dimensione^2) 0.21240 0.08412 2.525 0.019283 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 6.333 on 22 degrees of freedom
## Multiple R-squared: 0.892, Adjusted R-squared: 0.8822
## F-statistic: 90.9 on 2 and 22 DF, p-value: 2.321e-11 plot(processi ~ dimensione, data = dataset)
curve(coef(mod_quad)[1] + coef(mod_quad)[2] * x + coef(mod_quad)[3] * x ^ 2, from = min(dataset$processi), to = max(dataset$processi), add = TRUE, col = "blue")
curve(coef(mod_quad_2)[1] + coef(mod_quad_2)[2] * x + coef(mod_quad_2)[3] * x ^ 2, from = min(dataset$processi), to = max(dataset$processi), add = TRUE, col = "green")
par(mfrow = c(1,2))
plot(residuals(mod_quad_2) ~ dimensione[-outliers], data=dataset)
abline(h=0, col = "red")
qqPlot(residuals(mod_quad_2))
## 23 3
## 20 3 par(mfrow = c(1,1))6.3 Stima Cubica:
mod_cubica <- lm(processi ~ dimensione + I(dimensione ^ 3), data = dataset)
plot(processi ~ dimensione + I(dimensione ^ 2), data=dataset)
curve(coef(mod_cubica)[1] + coef(mod_cubica)[2] * x + coef(mod_cubica)[3] * x ^ 2, from = min(dataset$processi), to = max(dataset$processi), add = TRUE, col = "blue")
summary(mod_cubica)##
## Call:
## lm(formula = processi ~ dimensione + I(dimensione^3), data = dataset)
##
## Residuals:
## Min 1Q Median 3Q Max
## -27.450 -18.973 -9.846 3.653 70.223
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 67.385126 38.510468 1.750 0.092 .
## dimensione -3.056178 4.931776 -0.620 0.541
## I(dimensione^3) 0.002757 0.009430 0.292 0.772
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 28.43 on 26 degrees of freedom
## Multiple R-squared: 0.06926, Adjusted R-squared: -0.002335
## F-statistic: 0.9674 on 2 and 26 DF, p-value: 0.3933- Analisi con i residui:
par(mfrow = c(1,2))
plot(residuals(mod_cubica) ~ dimensione, data=dataset)
abline(h=0, col = "red")
qqPlot(residuals(mod_cubica))
## [1] 8 9 par(mfrow = c(1,1))- Analisi senza i residui:
mod_cubica_2 <- lm(processi ~ dimensione + I(dimensione ^ 3), data = dataset, subset=-outliers)
plot(processi ~ dimensione + I(dimensione ^ 3), data=dataset, subset= -outliers)
summary(mod_cubica_2)##
## Call:
## lm(formula = processi ~ dimensione + I(dimensione^3), data = dataset,
## subset = -outliers)
##
## Residuals:
## Min 1Q Median 3Q Max
## -14.3676 -3.3959 0.7948 3.9177 10.6041
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 95.263293 8.670800 10.987 2.12e-10 ***
## dimensione -6.684719 1.115278 -5.994 4.94e-06 ***
## I(dimensione^3) 0.005614 0.002124 2.643 0.0148 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 6.266 on 22 degrees of freedom
## Multiple R-squared: 0.8943, Adjusted R-squared: 0.8847
## F-statistic: 93.09 on 2 and 22 DF, p-value: 1.835e-11 plot(processi ~ dimensione, data = dataset)
curve(coef(mod_cubica)[1] + coef(mod_cubica)[2] * x + coef(mod_cubica)[3] * x ^ 2, from = min(dataset$processi), to = max(dataset$processi), add = TRUE, col = "blue")
curve(coef(mod_cubica_2)[1] + coef(mod_cubica_2)[2] * x + coef(mod_cubica_2)[3] * x ^ 2, from = min(dataset$processi), to = max(dataset$processi), add = TRUE, col = "green")
par(mfrow = c(1,2))
plot(residuals(mod_cubica_2) ~ dimensione[-outliers], data=dataset)
abline(h=0, col = "red")
qqPlot(residuals(mod_cubica_2))
## 23 3
## 20 3 par(mfrow = c(1,1))6.4 Stima logaritmica:
6.4.1 Per l’asse x :
mod_logaritmica_x <- lm(log(processi) ~ dimensione, data = dataset)
plot(log(processi) ~ dimensione, data=dataset)
abline(mod_logaritmica_x, col = "red")
summary(mod_logaritmica_x)##
## Call:
## lm(formula = log(processi) ~ dimensione, data = dataset)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.57997 -0.39783 -0.05473 0.10326 1.91984
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4.74800 0.54082 8.779 2.15e-09 ***
## dimensione -0.12045 0.03919 -3.073 0.0048 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.913 on 27 degrees of freedom
## Multiple R-squared: 0.2591, Adjusted R-squared: 0.2317
## F-statistic: 9.445 on 1 and 27 DF, p-value: 0.004798- Analisi con i residui
par(mfrow = c(1,2))
plot(residuals(mod_logaritmica_x) ~ dimensione, data=dataset)
abline(h=0, col = "red")
qqPlot(residuals(mod_logaritmica_x))
## [1] 6 9- Analisi senza i residui:
mod_logaritmica_x2 <- lm(log(processi) ~ dimensione , data = dataset, subset=-outliers)
plot(log(processi) ~ dimensione , data=dataset, subset= -outliers)
summary(mod_logaritmica_x2)##
## Call:
## lm(formula = log(processi) ~ dimensione, data = dataset, subset = -outliers)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.94733 -0.14401 0.08426 0.17129 0.64130
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 5.25862 0.30605 17.182 1.29e-14 ***
## dimensione -0.18396 0.02308 -7.972 4.56e-08 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.5047 on 23 degrees of freedom
## Multiple R-squared: 0.7342, Adjusted R-squared: 0.7227
## F-statistic: 63.54 on 1 and 23 DF, p-value: 4.56e-08 plot(log(processi) ~ dimensione, data = dataset)
curve(coef(mod_logaritmica_x)[1] + coef(mod_logaritmica_x)[2] * x + coef(mod_logaritmica_x)[3] * x ^ 2, from = min(dataset$processi), to = max(dataset$processi), add = TRUE, col = "blue")
curve(coef(mod_logaritmica_x2)[1] + coef(mod_logaritmica_x2)[2] * x + coef(mod_logaritmica_x2)[3] * x ^ 2, from = min(dataset$processi), to = max(dataset$processi), add = TRUE, col = "green")
abline(mod_logaritmica_x, col = "red")
par(mfrow = c(1,2))
plot(residuals(mod_logaritmica_x2) ~ dimensione[-outliers], data=dataset)
abline(h=0, col = "red")
qqPlot(residuals(mod_logaritmica_x2))
## 6 16
## 6 14 par(mfrow = c(1,1))6.4.2 Per l’asse y :
mod_logaritmica_y <- lm(processi ~ log(dimensione), data = dataset)
plot(processi ~ log(dimensione), data=dataset)
abline(mod_logaritmica_y, col = "red")
summary(mod_logaritmica_y)##
## Call:
## lm(formula = processi ~ log(dimensione), data = dataset)
##
## Residuals:
## Min 1Q Median 3Q Max
## -26.790 -18.949 -10.488 3.764 69.821
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 86.41 35.38 2.442 0.0214 *
## log(dimensione) -20.28 13.95 -1.454 0.1575
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 27.84 on 27 degrees of freedom
## Multiple R-squared: 0.07261, Adjusted R-squared: 0.03826
## F-statistic: 2.114 on 1 and 27 DF, p-value: 0.1575- Analisi con i residui
par(mfrow = c(1,2))
plot(residuals(mod_logaritmica_y) ~ dimensione, data=dataset)
abline(h=0, col = "red")
qqPlot(residuals(mod_logaritmica_y))
## [1] 8 9- Analisi senza i residui:
mod_logaritmica_y2 <- lm(processi ~ log(dimensione) , data = dataset, subset=-outliers)
plot(processi ~ log(dimensione), data=dataset, subset= -outliers)
summary(mod_logaritmica_y2)##
## Call:
## lm(formula = processi ~ log(dimensione), data = dataset, subset = -outliers)
##
## Residuals:
## Min 1Q Median 3Q Max
## -14.8395 -3.4371 0.2184 4.5486 11.1257
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 137.800 8.527 16.16 4.76e-14 ***
## log(dimensione) -45.185 3.426 -13.19 3.29e-12 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 6.443 on 23 degrees of freedom
## Multiple R-squared: 0.8832, Adjusted R-squared: 0.8781
## F-statistic: 173.9 on 1 and 23 DF, p-value: 3.289e-12 plot(processi ~ log(dimensione), data = dataset)
curve(coef(mod_logaritmica_y)[1] + coef(mod_logaritmica_y)[2] * x + coef(mod_logaritmica_y)[3] * x ^ 2, from = min(dataset$processi), to = max(dataset$processi), add = TRUE, col = "blue")
curve(coef(mod_logaritmica_y2)[1] + coef(mod_logaritmica_y2)[2] * x + coef(mod_logaritmica_y2)[3] * x ^ 2, from = min(dataset$processi), to = max(dataset$processi), add = TRUE, col = "green")
abline(mod_logaritmica_y, col = "red")
par(mfrow = c(1,2))
plot(residuals(mod_logaritmica_y2) ~ dimensione[-outliers], data=dataset)
abline(h=0, col = "red")
qqPlot(residuals(mod_logaritmica_y2))
## 23 3
## 20 3 par(mfrow = c(1,1))6.4.3 Per entrambi gli assi:
mod_logaritmica <- lm(log(processi) ~ log(dimensione), data = dataset)
plot(log(processi) ~ log(dimensione), data=dataset)
abline(mod_logaritmica, col = "red")
summary(mod_logaritmica)##
## Call:
## lm(formula = log(processi) ~ log(dimensione), data = dataset)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.6441 -0.4204 -0.0970 0.1637 1.8557
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 6.6319 1.1675 5.680 4.93e-06 ***
## log(dimensione) -1.3797 0.4603 -2.998 0.00578 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.9188 on 27 degrees of freedom
## Multiple R-squared: 0.2497, Adjusted R-squared: 0.2219
## F-statistic: 8.986 on 1 and 27 DF, p-value: 0.00578- Analisi con i residui
par(mfrow = c(1,2))
plot(residuals(mod_logaritmica) ~ dimensione, data=dataset)
abline(h=0, col = "red")
qqPlot(residuals(mod_logaritmica))
## [1] 6 9- Analisi senza i residui:
mod_logaritmica2 <- lm(log(processi) ~ log(dimensione) , data = dataset, subset=-outliers)
plot(log(processi) ~ log(dimensione), data=dataset, subset= -outliers)
summary(mod_logaritmica2)##
## Call:
## lm(formula = log(processi) ~ log(dimensione), data = dataset,
## subset = -outliers)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.05396 -0.14892 0.07807 0.25270 0.60856
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 8.1099 0.7047 11.508 5.07e-11 ***
## log(dimensione) -2.0952 0.2832 -7.399 1.59e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.5324 on 23 degrees of freedom
## Multiple R-squared: 0.7042, Adjusted R-squared: 0.6913
## F-statistic: 54.75 on 1 and 23 DF, p-value: 1.593e-07 plot(log(processi) ~ log(dimensione), data = dataset)
curve(coef(mod_logaritmica)[1] + coef(mod_logaritmica)[2] * x + coef(mod_logaritmica)[3] * x ^ 2, from = min(dataset$processi), to = max(dataset$processi), add = TRUE, col = "blue")
curve(coef(mod_logaritmica2)[1] + coef(mod_logaritmica2)[2] * x + coef(mod_logaritmica2)[3] * x ^ 2, from = min(dataset$processi), to = max(dataset$processi), add = TRUE, col = "green")
abline(mod_logaritmica, col = "red")
par(mfrow = c(1,2))
plot(residuals(mod_logaritmica2) ~ dimensione[-outliers], data=dataset)
abline(h=0, col = "red")
qqPlot(residuals(mod_logaritmica2))
## 6 23
## 6 20 par(mfrow = c(1,1))6.4.4 Unione delle tre tipologie più quella classica, quadratica e Cubica:
par(mfrow = c(2,2))
mod_logaritmica <- lm(log(processi) ~ log(dimensione), data = dataset)
plot(processi ~ dimensione, data=dataset)
plot(processi ~ dimensione + I(dimensione ^ 2), data=dataset)
plot(processi ~ dimensione + I(dimensione ^ 3), data=dataset)
plot(processi ~ log(dimensione), data=dataset)
plot(log(processi) ~ dimensione, data=dataset)
plot(log(processi) ~ log(dimensione), data=dataset)
par(mfrow = c(1,1))
summary(mod_logaritmica)##
## Call:
## lm(formula = log(processi) ~ log(dimensione), data = dataset)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.6441 -0.4204 -0.0970 0.1637 1.8557
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 6.6319 1.1675 5.680 4.93e-06 ***
## log(dimensione) -1.3797 0.4603 -2.998 0.00578 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.9188 on 27 degrees of freedom
## Multiple R-squared: 0.2497, Adjusted R-squared: 0.2219
## F-statistic: 8.986 on 1 and 27 DF, p-value: 0.005787 Calcolo dell’intervallo di Previsione
predict(mod2, newdata = data.frame(dimensione = 10), interval = "prediction") # dove 10 è la dimenisone che vogliamo controllare## fit lwr upr
## 1 36.30562 21.37121 51.24004CANCELLARE LE SEGUENTI RIGHE!
Qualche comando LaTeX utile per scrivere le formule
Formule nel testo: \(\sin(\pi x)\)
Formule su righe separate: \[ \sin(\pi x) \]
Momenti: \[ \mu_k, E(X^k), M_k, m_k, \bar{X}, \bar{x}, \sigma^2, Var(X), S^2 \] Stimatori e stime: \[ \hat \theta, \hat \mu, \hat \sigma^2, \hat \lambda \] \[ \tilde \theta, \tilde \mu, \tilde \sigma^2, \tilde \lambda \] Verosimiglianza: \[ L(\theta), \ell(\theta), \ell'(\theta), \ell''(\theta) \] Frazioni: \[ \frac{1}{\sqrt{2}} X_i, \frac{X_i}{Y_i} \]
Sommatorie: \[ \sum_{i=1}^n X_i, \sum_{i=1}^n \log X_i, \sum_{i=1}^n \sqrt{X_i}, \sum_{i=1}^n \frac{X_i}{Y_i} \] Integrali: \[ \int_{0}^1 f(x) dx, \int_{a}^{\infty} f(x) dx, \int_{-\infty}^{\infty} f(x) dx \]
Come scrivere i comandi R
2 * 3 / log(1.2)## [1] 32.90889sin(2 * pi / 20)## [1] 0.309017a <- 2 * c(1, 2)
a## [1] 2 4z <- qnorm(0.95)
z## [1] 1.644854t <- qt(0.95, df = 12)
t## [1] 1.782288